Dic₁₃₈₈₆ #72290199

order	55544
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₈₈₆

Splittings

Hall	C₅₃×C₁₃₁ ⋊ Q₈
Sylow	C₁₃₁ ⋊ Dic₁₀₆
Sylow	C₅₃ ⋊ Dic₂₆₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₉₄₃
commutator, socle, prime radical, cosocle	C₂×C₅₃×C₁₃₁	C₂²
Fitting	C₄×C₅₃×C₁₃₁	C₂
hypercenter	C₄	D₆₉₄₃
hypocenter	C₅₃×C₁₃₁	Q₈
LCS	G ▷ C₂×C₅₃×C₁₃₁ ▷ C₅₃×C₁₃₁
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 192
transitive degree	μ_t(G) = 55544

A minimal faithful permutation representation — 2 generators on 192 points:

(1,2,3,4)(5,8,7,6)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61)(62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192);(1,5,3,7)(2,6,4,8)(10,61)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(63,192)(64,191)(65,190)(66,189)(67,188)(68,187)(69,186)(70,185)(71,184)(72,183)(73,182)(74,181)(75,180)(76,179)(77,178)(78,177)(79,176)(80,175)(81,174)(82,173)(83,172)(84,171)(85,170)(86,169)(87,168)(88,167)(89,166)(90,165)(91,164)(92,163)(93,162)(94,161)(95,160)(96,159)(97,158)(98,157)(99,156)(100,155)(101,154)(102,153)(103,152)(104,151)(105,150)(106,149)(107,148)(108,147)(109,146)(110,145)(111,144)(112,143)(113,142)(114,141)(115,140)(116,139)(117,138)(118,137)(119,136)(120,135)(121,134)(122,133)(123,132)(124,131)(125,130)(126,129)(127,128)

Polycyclic presentation:

2^3,53,131|a^2=c,b^2=c,b^a=bc,d^a=d^52,e^a=e^130

Conjugacy classes & character table

13889 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 53, 131]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^52);;
gap> SetConjugate(coll, 5, 1, F.5^130);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13886 #72290199